The idea of approximation of infinite structures by means of finite or compact objects is prevalent in modern mathematics. It is the aim of this project to gain insight into the structure of infinite groups and non-compact manifolds by means of sofic approximations.
The project focuses on studying geometric properties for sofic approximations of discrete groups and manifolds. A group \(\Gamma\) is sofic if it is possible to find local models in symmetric groups \(Sym(X)\) that are almost multiplicative with respect to a certain metric. Similarly, a cocompact \(\Gamma\)-manifold \(M\) is called sofic if there exist compact manifolds that locally look like \(M\) more and more with high probability. A countable collection of finite sets that witness stronger and stronger approximations for an exhaustion of the group \(\Gamma\) is a sofic approximation - similarly for manifolds.
Sofic groups were introduced by Gromov in his work on Gottschalk's surjunctivity conjecture and later by Weiss. Since then they have played a fundamental role in research in dynamical systems and their connections to \(L^2\)-invariants, notably being the largest class of groups for which the concept of entropy is well defined and the notion of mean dimension has been extended as well as revealing a strong approximation property in the context of \(L^2\)-invariants.
This project consists of three interacting themes:
i) Sofic boundary actions
ii) Study of analytic/geometric properties of groups using actions at infinity
iii) Sofic manifolds